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An illustration of Stokes' theorem, with surface Σ, its boundary ∂Σ and the normal vector n.The direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule (i.e., the right hand the fingers circulate along ∂Σ and the thumb is directed along n).
In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or , and the divergence theorem is the case of a volume in . [2] Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.
By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, [4] = = = Here, the closed integration path ∂S is the boundary or perimeter of an open surface S , whose infinitesimal element normal d S = n dS is oriented according to the right-hand rule .
Stokes' theorem. It is named after Sir George Gabriel Stokes (1819–1903), although the first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes. The theorem acquired its name from Stokes' habit of including it in the Cambridge prize examinations. In 1854 he asked his students to prove ...
In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field.
A tensor form of a vector integral theorem may be obtained by replacing the vector (or one of them) by a tensor, provided that the vector is first made to appear only as the right-most vector of each integrand. For example, Stokes' theorem becomes
In order to apply this to the Navier–Stokes equations, three assumptions were made by Stokes: The stress tensor is a linear function of the strain rate tensor or equivalently the velocity gradient. The fluid is isotropic. For a fluid at rest, ∇ ⋅ τ must be zero (so that hydrostatic pressure results).
The second term on the right hand side is the displacement current as originally conceived by Maxwell, associated with the polarization of the individual molecules of the dielectric material. Maxwell's original explanation for displacement current focused upon the situation that occurs in dielectric media.